Your perception of 'slightly obsequious, too present service' is really more of a You Problem than you may realise, as a British person.
People are not fawning over you, either as part of some false act or just because they think you'll give a bigger tip. If they're working as servers, it's because they chose to do that, they're outgoing people, they like the hours, and they're happy to maximize their income by being cheerfully attentive. There are options besides restaurant work for people in the US. If they worked as cashiers in a supermarket, no amount of cheeriness or efficiency would increase their income.
If you want less of the 'too present' bit, simply tell them you'll signal to them when you need them, and you're fine otherwise. You can ask directly for what you want. Nobody will be miffed.
And they are paid properly if you give the normal tip.
If you think of the servers as independent operators within the restaurant environment, it might make the concept of dividing your payments easier to understand. The listed menunproce of the food is one portion of what you pay. The service is the other portion.
The servers and other front of house staff are paid a basic wage which the servers then top up by the portion of the bill that the patrons pay directly to them in tips, cutting out the middle man (the owner/ employer).